By any normal reckoning, the French left should be on the verge of an electoral breakthrough. Nicolas Sarkozy is the most unpopular president in the 50-odd-year history of France's fifth republic. Damned as the bling-bling "president of the rich" who championed the market on the eve of its greatest crisis in the postwar era, barely one in five think he's doing a good job.

There is even speculation that he may not stand for a second term. And whatever bounce the embattled president might derive from his new role as a war leader in the Libyan conflict is thought unlikely to last. But although the main opposition Socialist party led the field in countrywide local elections last Sunday with 25% of the vote, while Sarkozy's centre-right Union for a Popular Movement scored a humiliating 17%, it was the far-right National Front that registered the strongest advance, coming in less than two percentage points behind the ruling party. Already the Sarkozy camp is in disarray about how to respond to the front's success, having failed to win back supporters with attacks on multiculturalism, an anti-Roma migrant campaign and a legal ban on the Islamic face veil. In some polls, the front's new leader, Marine Le Pen, was ranked second for the crucial runoff in next year's presidential elections.

But Le Pen has rebuilt the front's fortunes not only on the back of anti-immigration and Islamophobic incitement at a time of record youth unemployment, insecurity and squeezed living standards. She has also been stealing the left's clothes on public services, social protection, neoliberal globalisation and the "ultra-liberal ideology of financial capitalism", even invoking the names of historic French communist leaders such as Maurice Thorez to appeal to working-class voters alienated from what they see as a Tweedledum-Tweedledee political establishment.

Such a pitch is aimed at mining a deep-seated radical strain in French public opinion. For example, in a recent poll 43% said they thought free-market capitalism was "fatally flawed" and should be replaced with a different economic system, more than in any other country surveyed – and six out of 10 said this month they would like to see a "revolt" over social and economic problems. It's that kind of sentiment that has been repeatedly mobilised by the left on the streets, most recently in last autumn's wave of strikes and mass protests against Sarkozy's pension reforms. It also fuelled the successful 2005 No referendum campaign against the market-oriented European constitution, which drew heavily on the opposition of the left – though not the socialist leadership.

But turning that into electoral success, or radical reform when the left has been in office, has been another matter. As Annick Coupé of the leftist Solidaires trade unions puts it: "There is a very strong feeling of injustice, but people don't see a clear alternative."

Like New Labour in Britain, the French socialists are widely regarded as having implemented similar neoliberal policies to the parties of the right, such as privatisation and European liberalisation. Polls show more than two-thirds of French voters believe the mainstream left and right have become increasingly alike. That has been seized on by the smaller parties of the left and fed into the battle to be next year's Socialist presidential candidate. But paradoxically, the favourite is now Dominique Strauss-Kahn, managing director of the International Monetary Fund, architect of the French Socialist privatisation programme in the late 1990s, Blair-like champion of globalisation, friend of wealthy media proprietors and the epitome of the cosy political establishment that so alienates the French public.

The polls

Current polling nevertheless shows DSK — as he is known in France — beating Sarkozy by 61% to 39% if he stands. Part of the attraction seems to be his presidential style, and part the silence induced by his IMF role that protects him from the mud-wrestling of French politics. But his support also appears to feed on itself. The experience of 2002, when Le Pen's father beat the Socialist candidate in the first round, has led many voters determined to get rid of Sarkozy to back whichever Socialist is polling most strongly in a weak field.

Strauss-Kahn seems certain to tack left if the time comes. But whether his reputation will survive the party's new US-style primaries – expected to include Ségolène Royal, who lost to Sarkozy in 2007, her ex-husband François Hollande and perhaps the more traditional Martine Aubry – is another question. And even the Socialist party's spokesman, Benoit Hamon, told the Guardian that Strauss-Kahn had failed to "sign up to the rejection of neoliberalism" necessary in the wake of the crisis.

The more radical wing of the French left – which together with the Greens potentially commands more than 20% of the vote and could exert a powerful pressure on the Socialists to give voice to the scale of French social discontent — is beset by its own weaknesses. The shrunken Communist party, which still has thousands of elected representatives, is part of a Left Front with the outspoken former Socialist senator Jean-Luc Mélenchon that polled 9% last Sunday. But the New Anti-Capitalist party (NPA), formed two years ago by a charismatic postman, Olivier Besancenot, whose poll ratings at one time outstripped those of the Socialist leaders, has since haemorrhaged members and support. Partly it has been damaged by its refusal to support Socialists in the crucial second runoff round of the French voting system.

But the party has also been racked by divisions over secularism and Islam. Its initial endorsement of a young headscarf-wearing Muslim woman, Ilham Moussaid, as a local candidate triggered internal uproar. Controversy over the hijab and the interpretation of the French secular tradition – particularly when it is being used as a stick to beat an ethnic and religious minority, rather than target a rightwing Catholic establishment – has cut like a knife throughout the French left. Last month the NPA voted to ban women who wear the hijab from representing the party and Moussaid, among others, has walked out.

The possibility that despite Sarkozy's unprecedented unpopularity, a weakly-led and divided left could still miss an open goal is taken seriously. Hamon warns: "Sarkozy can't win the presidency – but we can still lose it." Perhaps more likely, as the editor of the influential left-leaning Le Monde Diplomatique, Serge Halimi, puts it: "The paradox of French politics is that people clearly want a shift to the left in economic policy. But because of the way the electoral system works they may end up with a choice between Strauss-Kahn and Nicolas Sarkozy – the two men of globalisation that the French hate".